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Unformatted text preview: STT 861: Fall 2007 1 Midterm 1 1. Let A 1 ,A 2 , ,A n be events defined on the same sample space. (a) Prove that P ( A 1 A 2 ) P ( A 1 ) + P ( A 2 ) 1. (b) Prove that P ( n i =1 A i ) n i =1 P ( A i ) ( n 1). 2. Suppose that events A 1 ,A 2 ,A 3 and A 4 are mutually independent. Show that for any i and j , i negationslash = j , A c i and A c j are independent. ( Hint: show that P ( A c i A c j ) = P ( A c i ) P ( A c j ) ) 3. Consider a vector X = ( x 1 ,x 2 ,x 3 ) Z 3 , where Z is the set of all integers. (a) How many vectors X are in Z 3 with positive integer components x i , i = 1 , 2 , 3 that sum to seven? (b) How many vectors X are in Z 3 with positive odd integer components x i , i = 1 , 2 , 3 that sum to nine? 4. There are 21 students in the class STT861 section 2. Female students are 12 among 21 students. They decide to choose a representative of the class and a secretary of the representative. (a) What is the probability that they choose students with different gender for their representative and secretary?...
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